Let Q(x,y)=0 be an hyperbola in the plane. Given real numbers $��\equiv��^{2n)}=\{��_{ij}\}_{i,j\geq0,i+j\leq2n}$, with $��_{00}>0$, the truncated Q-hyperbolic moment problem for ��entails finding necessary and sufficient conditions for the existence of a positive Borel measure ��, supported in Q(x,y)=0, such that $��_{ij}=\int y^{i}x^{j} d��(0\leq i+j\leq2n)$. We prove that ��admits a Q-representing measure ��(as above) if and only if the associated moment matrix $\mathcal{M}(n)(��)$ is positive semidefinite, recursively generated, has a column relation Q(X,Y)=0, and the algebraic variety $\mathcal{V}(��)$ associated to ��satisfies $card\mathcal{V}(��)\geq\rank\mathcal{M}(n)(��)$. In this case, $rank\mathcal{M}(n)\leq2n+1$; if $rank\mathcal{M}(n)\leq2n$, then ��admits a $rank\mathcal{M}(n)$-atomic (minimal) Q-representing measure; if $rank\mathcal{M}(n)=2n+1$, then ��admits a Q-representing measure ��satisfying $2n+1\leqcard supp��\leq2n+2$.